Consider the $N^2$ terms of the form $g[m]h[n], 0\leq m < N, 0 \leq n < N$.
Each term occurs once and only once in the $2N-1$ sums that define the values of $yL[k]$, $0 \leq k < 2N$.
Each term occurs once and only once in the $N$ sums that define the values of $yC[\ell], 0 \leq \ell < N$.
So, make a $N\times N$ table of values $(m,n)$ and figure out which
$k$ and which $\ell$ each $(m,n)$ maps to. Write the answers in matrix form where the $(m,n)^{\text{th}}$ entry is of the form $k~ ||~ \ell$. Then ponder the questions:
- For a fixed value of $\ell$, what are the various values of $k$ that you have listed in all matrix entries of the form $k~ ||~ \ell$?
- Are these values of $k$ associated with any other values of $\ell$?
It is sincerely to be hoped that you will find the answer to the second question to be No. If so, and if the answers to the first question are "$k$ can equal $k_1 k_2, \cdots, k_i$", then you have just discovered that $yC[\ell] = yL[k_1]+yL[k_2]+ \cdots + yL[k_i]$.